Michael Reid wrote: <<

Let N = 2K+1 for some positive integer K. Evaluate explicitly

f(N) := Product (5 - 4*cos(2*j*pi/N))

where j runs through 1,2,...,K, and prove your answer.

over R , the polynomial x^(2K) + x^(2K-1) + ... + x^2 + x + 1 factors as Product (x^2 - 2 cos(2 j pi / (2K + 1)) x + 1) , the product being for j = 1, 2, ... , K . this is easy to see by factoring the polynomial completely over C , and then pairing up complex conjugate factors. now take x = 2 to get the answer: 2^(2K+1) - 1 .

...

Yes, exactly my reasoning. (((This came up when I needed to calculate the norm N(2-w) of 2-w for w = exp(2pi*i/p), p an odd prime, in the cyclotomic ring Z[w]. By definition in this case, N(w) = (2-w)(2-w^2)...(2-w^(p-1)), so I naively paired up the inverses and calculated a few cases by hand: p=3 -> N(2-w) = 7; p=5 -> N(2-w) = 31; p=7 -> N(2-w) = 127 . . . a clear pattern emerging. But pairing up the inverses only obscured the reason for this, giving factors of the form (*) (5 - 4 cos(2pi*k/p)) = (2-w^k)(2-w^(-k)). By *not* pairing the inverses one gets, as Michael did, that since (X-w)(X-w^2)...(X-w^(p-1)) = (X^p - 1)/(X-1), setting X = 2 gives N(w) = 2^p - 1. This would've taken me much longer had I not known (*) to begin with.))) --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele